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\begin{document}

\preprint{}

\title{Quantum Field Theory Exceptions to Uncertainty Bounds from Quantum Mechanics}
 
\pacs{}

\author{Cael L.~Hasse}
\email{Electronic address: cael.hasse@adelaide.edu.au}
\affiliation{Special Research Centre for the Subatomic Structure of Matter and Department of Physics, University of Adelaide 5005, Australia.}
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\date{\today}

\begin{abstract}
We outline a method of constructing states of many fermions that are simultaneous eigenstates of incompatible observables; non-commuting occupation numbers for incompatible modes.  The method shows the existence of states where every particle has definite values for any two quantum numbers of the particles.  Thus the uncertainty bounds on single particle states cannot be extrapolated to many particle fermion states.  In particular, we construct a state such that every particle has a definite momentum and position.  This effect occurs with a loss of reference frame for quantum numbers of indistinguishable particles.  A basis ambiguity is then admitted upon this loss of reference frame, allowing for a basis dependent mixing of properties between the particles.
\end{abstract}

\maketitle

An aspect of fundamental importance in quantum theory is our inability to predict the outcome of every possible measurement of any state; even in principle \cite{Heis27}.  This observation was given the name the Uncertainty Principle \cite{Rob29} and played a pivotal role in the conceptual clarification of quantum theory.  It also plays an important role in quantum information theory and quantum cryptography \cite{Dam05,Wehn10,Bart10,Oppen10}.  In this letter we discuss the manifestation of the uncertainty principle in quantum field theory.  For example, Bohr \cite{Bohr33} used uncertainty bounds between different components of the electromagnetic field to analyse its measurability in quantum electrodynamics.  We shall focus instead on fermionic fields, and in particular their occupation numbers rather than the field operators themselves.

The uncertainty bound between two observables in its original formulation \cite{Heis27}, is directly related to the non-commutativity of those observables.  For observables $A$ and $B$, 
%
\begin{equation}
\Delta A \Delta B \geq \tfrac{1}{2} |\langle \psi | [A,B]|\psi\rangle|\,, \label{gup}
\end{equation}
%
where $\Delta A$ and $\Delta B$ are the standard deviations of $A$ and $B$ respectively.  There are limitations to this formulation \cite{Deu83} namely; the bound can only be non-zero for observables with an infinite range of eigenvalues and the bound is generally dependent on the state $|\psi\rangle$.

Relation (\ref{gup}) is most useful when the commutator is proportional to the identity such that the right hand side becomes independent of the state.  The bound then becomes a statement of the fundamental limitations quantum mechanics imposes on a particular theory.  Canonically conjugate observables in particular have this form.  In general however, non-commutativity only implies the non-existence of a {\it complete} set of simultaneous eigenstates.  An incomplete set of simultaneous eigenstates can still exist.  If this set does exist, then elements of this set give zero for the right hand side of (\ref{gup}).

Importantly, the observables are defined within particular classes of theory.  For instance the position and momentum of a particle in wave mechanics are canonically conjugate such that there exists a fundamental minimum uncertainty with the pair of observables.  In quantum field theory however, the conjugate variables are not the position and momentum of the particles.  Moreover, the indistinguishability of identical particles imposes a superselection rule on all observables, namely; the extra constraint that they are symmetric under permutations of the particles.  The relevant observables then become occupation numbers.  For instance, position and momenta densities, which are not conjugate.  We shall now show that in the case of fermions, simultaneous eigenstates of non-commuting occupation number operators exist and discuss the physical interpretation of such states.

Suppose we have fermion creation operators $\adag_i,\,1\leq i \leq n$, where $i$ labels distinct states with a set of quantum numbers.  Also suppose we have some state with the factor $\adag_1\dots\adag_n$.  We can apply a passive Bogoliubov transformation to these ladder operators of the form
%
\begin{equation}
\abadag_i := \sum_{j=1}^{n} U_{ij} \adag_j\,, \label{bog}
\end{equation}
%
where $U$ is a unitary matrix, to get new ladder operators;
%
\begin{equation}
[\aba_i,\abadag_k]_{+} = \delta_{ik}, \qquad [\aba_i,\aba_k]_{+} = 0 = [\abadag_i, \abadag_k]_{+}\,.
\end{equation}
%
These new ladder operators create particles in superpositions of the states created by the $\abadag_j$s.  The original factor we are considering can be related to these new particles:
%
\begin{align}
\adag_1\dots\adag_n &= \sum_{i_1,\dots,i_n} U^{-1}_{1i_1}\dots U^{-1}_{ni_n} \abadag_{i_1}\dots\abadag_{i_n} \notag \\
%
&= \sum_{i_1,\dots,i_n} U^{-1}_{1i_1}\dots U^{-1}_{ni_n} \epsilon_{i_1\dots i_n} \abadag_1\dots\abadag_n \notag \\
%
&= \det(U^{-1}) \abadag_1\dots\abadag_n\,.
\end{align}
%
Note $\det(U^{-1})$ is a phase.  Let us now focus on this factor acting on the Fock vacuum $|0\rangle$ (which is invariant under Bogoliubov transformations that do not mix creation and annihilation operators),
%
\begin{equation}
|\phi\rangle := \adag_1\dots\adag_n |0\rangle = \det(U^{-1})\abadag_1\dots\abadag_n |0\rangle\,.
\end{equation}
%
The occupation number operators for individual modes are given by
%
\begin{equation}
\eta_k := \adag_k a_k\quad \mbox{and}\quad \eba_k := \abadag_k\aba_k\,.
\end{equation}
%
One can see that $|\phi\rangle$ is a simultaneous eigenstate of $\eta_k$ and $\eba_j$ for all $1\leq j,k\leq n$.  This is despite the observables not commuting,
%
\begin{equation}
[\eta_i,\eba_j] = U_{ji}\adag_i\aba_j - U^{*}_{ji}\abadag_j a_i\,.
\end{equation}
%
These occupation number operators can refer to modes where single particle states or multi-particle distinguishable particle states have  bound net uncertainty, given by an uncertainty relation.  For example, we can look at position and momentum.

Consider a $3$-dimensional spatial lattice with side lengths $L$ and periodic boundary conditions.  With lattice spacing $s,\, \vec{x} = \vec{n}s$ where $n=1,\dots,N$ such that $Ns=L$ and lattice momenta $\vec{k} = 2\pi\vec{m}/L$, with $m_i = 1,\dots,N$.  Define $\adag_{\vec{r}q}$ to create a fermion with lattice momentum $\vec{k}_r = 2\pi\vec{r}/L$ $(r_i = 1,\dots,N)$ and all other quantum numbers given by $q$, which we define as independent of $\vec{r}$.  Further define
%
\begin{equation}
\pdag_{\vec{x}q} = \frac{1}{N^{3/2}} \sum_{r_1,r_2,r_3=1}^{N} e^{-i\vec{x}\cdot\vec{k}_r} \adag_{\vec{r}q}\,.
\end{equation}
%
In the continuum limit, this operator creates a particle at a definite position $\vec{x}$ with other quantum numbers given by $q$.  The ladder operators $\adag_{\vec{r}q}$ and $\pdag_{\vec{x}q}$ are related by the sort of Bogoliubov transformation we have been considering.  Thus we can examine the state
%
\begin{equation}
|\phi'\rangle = \prod_{r_1,r_2,r_3=1}^{N} \adag_{\vec{r}q} |0\rangle = (\mbox{phase}) \prod_{n_1,n_2,n_3=1}^{N} \pdag_{\vec{x}(\vec{n})q} |0\rangle\,,
\end{equation}
%
where the ordering is arbitrary.

Let $\eta_{\vec{r}q} = \adag_{\vec{r}q}a_{\vec{r}q}$ and $\eba_{\vec{x}q} = \pdag_{\vec{x}q}\psi_{\vec{x}q}$.  Then
%
\begin{equation}
\eta_{\vec{r}q} |\phi'\rangle = |\phi'\rangle = \eba_{\vec{x}q} |\phi'\rangle \quad \forall\, \vec{r}\,\,\mbox{and}\,\,\vec{x}\,.
\end{equation}
%
We can conclude from this that, in the continuum limit, every particle in $|\phi'\rangle$ has definite momentum {\it and} every particle has a definite position.  

Some care must be taken as to the description of this state.  The Fock spaces for incompatible modes have {\it different} tensor product structures.  The definition of a subsystem (i.e. a particle) then becomes {\it relative} to the type of modes considered \cite{Zan04}.  Thus we cannot say that {\it each} particle in $|\phi'\rangle$ has definite momentum and position.  In this situation there are multiple compatible pictures; one for instance where every particle has a definite momentum while another being where every particle has a definite position.

Moreover, this effect can occur for any `incompatible' quantum numbers of fermions.

As another example, consider a dineutron state where both neutrons are in an $s$-wave and their spins are anti-aligned in some $z$-direction, given by
%
\begin{equation}
|\gamma\rangle := \adag_{N1}\adag_{N2} |0\rangle\,,
\end{equation}
%
where $\adag_{N1}$ and $\adag_{N2}$ create's $s$-wave neutrons with spin up and down respectively. 

Similarly to the previous example, the spin quantum numbers can be seen to have definite values in {\it any} direction.  Let $U_{ij} = (\exp\{i\vec{\theta}\cdot\vec{\sigma}\})_{ij}$, where $\vec{\sigma} = (\sigma_1,\sigma_2,\sigma_3)$ is the Pauli matrix vector, such that $\abadag_{N\vec{\theta_1}}=\sum_{j=1}^{2}(\exp\{i\vec{\theta}\cdot\vec{\sigma}\})_{ij}\adag_{Nj}$.  Thus,
%
\begin{equation}
|\gamma\rangle = \abadag_{N\vec{\theta}_1}\abadag_{N\vec{\theta}_2}|0\rangle \qquad \forall\, \vec{\theta}\,.
\end{equation}

It is important that all quantum numbers not considered,\footnote{Such as the ones labelled by $q$ in $|\phi'\rangle$ and the isospin and angular momenta (and any others) in $|\gamma\rangle$.}, were the same for each particle in the Bogoliubov transformation.  In principle, one may consider combinations of quantum numbers and superpositions thereof, though this may be difficult in practice.

For indistinguishable particles, quantum numbers become {\it reference frames} \cite{Bart07} for the other quantum numbers of the particle.  Take for instance two indistinguishable particles with quantum numbers spin and position.  Suppose one whose field has compact support in region $\mathcal{A}$ has spin up in some $z$-direction and the other has compact support in a disjoint region $\mathcal{B}$ with spin down.  If we ask the question, which particle is in region $\mathcal{A}$?  The physically meaningful answer relates to the observable qualities of the particle, ie., the particle in region $\mathcal{A}$ is the one whose spin is up.  If both particles are spin up (such that $q$ is the same for both) then we have lost the spin reference frame to identify which particle is in that region.

This loss of reference frame then admits a basis ambiguity.  We could for example write this two particle state in a {\it non-Fock} representation where the (anti-) symmetry is manifest;
%
\begin{equation}
|\omega\rangle := |\psi_A\rangle_1 |\uparrow\rangle_{1_s} |\psi_B\rangle_2 |\uparrow\rangle_{2_s} \pm |\psi_B\rangle_1 |\uparrow\rangle_{1_s} |\psi_A\rangle_2 |\uparrow\rangle_{2_s}\,,
\end{equation}
%
where $\psi_{A/B}$ label fields with compact support in region $A/B$ and the relative phase $+$ and $-$ is for bosons and fermions respectively.  As the spins (or the polarizations) are the same for both particles, these quantum numbers can be factorized;
%
\begin{equation}
|\omega\rangle = |\psi\rangle|\uparrow\rangle_{1_s}|\uparrow\rangle_{2_s}
\end{equation}
%
The factor 
%
\begin{equation}
|\psi\rangle = |\psi_A\rangle_1 |\psi_B\rangle_2 \pm |\psi_B\rangle_1 |\psi_A\rangle_2 \in \mathcal{H}_1\otimes\mathcal{H}_2\,,
\end{equation}
%
is now the form of a Bell state with its associated basis ambiguity (a change of basis being analogous to our passive Bogoliubov transformation).  However, due to the permutation symmetry of the observables, no orthogonal bases of the subsystem Hilbert spaces ($\mathcal{H}_1$ or $\mathcal{H}_2$) in this representation correspond to eigenstates of any observables.  Thus the basis ambiguity of of $|\psi\rangle$ has a {\it different} physical manifestation to that normally considered in Bell states, allowing for instance the effect considered in this paper.

The general case of the situation is exemplified by the state,
%
\begin{equation}
\prod_{i} \adag_{iq} |0\rangle\,,
\end{equation}
%
where $\adag_{iq}$ creates a particle with orthonormal modes labelled by $i$ and all other quantum numbers labelled by $q$.  It is situations of this form where, in the case of fermions, uncertainty bounds on quantum numbers of single particles can be sidestepped.

In this letter we have considered the kinematics of quantum field theory and come to the counter-intuitive notion that a loss of reference frame can be used to sidestep an apparent fundamental limitation of quantum mechanics.  However, we have not considered dynamics.  The construction, stability, and manipulation of such states in the laboratory may prove difficult.

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\bibitem{Dam05} I.~Damgaard, S.~Fehr, L.~Salvail, C.~Schaffner, {\it in Cryptography in the bounded quantum-storage model}, FOCS.\ 46th Annual IEEE Symposium on Foundations of Computer Science, pp.\ 449–458. IEEE Computer Society Press, Los Alamitos (2005).

\bibitem{Wehn10} S.~Wehner, A.~Winter, N.\ J.\ Phys.\ \textbf{12}, 025009 (2010).

\bibitem{Bart10} M.~Bartam M.~Christandl, R.~Colbeck, J.~M.~Renes, R.~Renner, Nature \textbf{6}, 659 (2010).

\bibitem{Oppen10} J.~Oppenheim, S.~Wehner, Science \textbf{330}, 1072 (2010).

\bibitem{Bohr33} N.~Bohr, in {\it Niels Bohr Collected Works}, edited by J.~Kalckar (Elsevier, Amsterdam, 1996), Vol.\ 7, p.\ 55.

\bibitem{Deu83} D.~Deutsch, Phys.\ Rev.\ Lett.\ \textbf{50}, 631 (1983).

\bibitem{Zan04} P.~Zanardi, D.~A.~Lidar and S.~Lloyd, Phys.\ Rev.\ Lett.\ \textbf{92}, 6 (2004).

\bibitem{Bart07} S.~O.~Bartlett, T.~Rudolf, and R.~W.~Spekkens, Rev.\ Mod.\ Phys.\ \textbf{79}, 555 (2007).

\end{thebibliography}

\end{document}

